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The following integral,

$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$

is clearly positive, which proves that $\pi < 22/7$.

Is there a similar integral which proves $\pi > 333/106$?

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Hey, how do you know that it is clearly positive? is there something about the integral that makes it positive? – Tyler Hilton Aug 10 '10 at 20:01
If $f(x)>0$ then $$\int\limits_{a}^{0} f(x)>0$$ – anonymous Aug 10 '10 at 20:06
@Affan, you're integrating a fraction of even powers: it can't be negative. – Andrea Ambu Aug 11 '10 at 7:13
You can use $\pi = \int_0^1 \frac{4}{1+x^2}$. Now the power series of $\frac{4}{1+x^2}$ is alternating, thus stopping after odd/even numbers of terms gives under and overevaluations. Thus, for all $a< \pi <b$ you can find $m, n$ so that $a< \int_0^1 P_m(x) dx < \pi < \int_0^1 P_n(x) < b$, where $P_n$ denotes a Taylor Polynomial. – N. S. Aug 3 '12 at 13:25

3 Answers 3

up vote 70 down vote accepted

This integral would do the job:

$$\int_0^1 \frac{x^5(1-x)^6(197+462x^2)}{530(1+x^2)}= \pi -\frac{333}{106}$$

  • Also you can refer to S.K. Lucas Integral proofs that $355/113 > \pi$, Gazette, Aust. Math. Soc. 32 (2005), 263-266.

  • This is the link. (Thanks to lhf for pointing out.)

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Great! I had not expected anyone to answer this so quickly! – anon Aug 9 '10 at 21:10
How does one come up with that integral, may I ask? – ShreevatsaR Aug 9 '10 at 21:33
5 Please read this article – anonymous Aug 9 '10 at 21:38
The link above is broken. The current one is – lhf Jun 10 '11 at 19:05
@lhf: Thanks for the link. – user9413 Jun 10 '11 at 19:09

Although this is not exactly an answer to the question, it seems sufficiently related to mention: there are some direct generalizations, given on the Wikipedia page about this integral. For instance, $$0 < \frac14\int_0^1\frac{x^8(1-x)^8}{1+x^2}\ dx=\pi -\frac{47171}{15015}$$

In general, $$\frac1{2^{2n-1}}\int_0^1 x^{4n}(1-x)^{4n}\ dx <\frac1{2^{2n-2}}\int_0^1\frac{x^{4n}(1-x)^{4n}}{1+x^2}\ dx <\frac1{2^{2n-2}}\int_0^1 x^{4n}(1-x)^{4n}\ dx$$

which for $n=1$ (the integral in the question) gives slightly better bounds than just $\pi < 22/7$: $$ \frac{1}{1260} < \frac{22}{7} - \pi < \frac{1}{630}$$

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Thank you! Doesn't that imply that pi is irrational? – anon Aug 9 '10 at 21:17
@muad: Without checking the asymptotics of the respective numerators and denominators, I'm not sure. It doesn't follow simply from the fact that the left and right expressions go to 0: for instance 1 is rational, but you could still find sequences of rational numbers $x_n, y_n, z_n$ such that $x_n$ and $y_n$ go to 0, but $x_n < 1 - z_n < y_n$ for all $n$. – ShreevatsaR Aug 9 '10 at 21:31
@anon For this to work, you'd have to show that the rational approximants converged suitably fast. See Dirichlet's irrationality test. – A Walker Oct 19 '11 at 6:23

In the beginning of 2009 I was posting re similar issue at several sites, namely, at sci.math.symbolic,, etc.

To repeat: In Paper 1 Lucas found, by brute-force search using Maple programming, several different variants of integral identities which relate each of several first Pi convergents (described in terms of OEIS sequences as A002485(n)/A002486(n)) to Pi.

Further, in my above-mentioned postings, I conjectured the following identity below, which represents a generalization of Stephen Lucas' experimentally obtained identities between Pi and its convergents:

$$(-1)^n\cdot(\pi - \text{A002485}(n)/\text{A002486}(n))$$

$$=(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^m(k+(i+k)x^2)\big)/(1+x^2)\; dx$$

where integer n = 0,1,2,3,... serves as index for terms in OEIS A002485(n) and A002486(n), and {i, j, k, l, m} are some integers (to be found experimentally or otherwise), which are probably some functions of n.

The "interesting" (I think) part of my generalization conjecture is that "i" is present in both:

denominator of the coefficient in front of the integral and in the body of the integral itself

For example for $\frac{22}{7}$

$$\frac{22}{7} - \pi = \int_{0}^{1}\frac{x^4(1-x)^4}{1+x^2}\,\mathrm{d}x$$

with $n=3, i=-1, j=0, k=1, l=4, m=4$ - with regards to my above suggested generalization.

In Maple notation

i:=-1; j:=0; k:=1; l:=4; m:=4;Int(x^l*(1-x)^m*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 22/7 - Pi

It also works for found by Lucas

formula for $\frac{333}{106}$

$$\pi - \frac{333}{106} = \frac{1}{530}\int_{0}^{1}\frac{x^5(1-x)^6(197+462x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=4, i=265, j=1, k=197, l=5, m=6$ -with regards to my above suggested generalization.

In Maple notation i:=265; j:=1; k:=197; l:=5; m:=6;Int(x^l*(1-x)^m*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields Pi - 333/106

And it works for Lucas's formula for $\frac{355}{113}$

$$\frac{355}{113} - \pi = \frac{1}{3164}\int_{0}^{1}\frac{(x^8(1-x)^8(25+816x^2)}{(1+x^2)}$$

with $n=5, i=791, j=2, k=25, l=8, m=8$ -with regards to my above suggested generalization.

In Maple notation

i:=791; j:=2; k:=25; l:=8; m:=8;Int(x^m*(1-x)^l*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 355/113 - Pi

And it works as well for Lucas's formula for $\frac{103993}{33102}$

$$\pi - \frac{103993}{33102} = \frac{1}{755216}\int_{0}^{1}\frac{x^{14}(1-x)^{12}(124360+77159x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=6, i= -47201, j=4, k=124360, l=14, m=12$ -with regards to my above suggested generalization.

In Maple notation

i:=-47201; j:=4; k:=124360; l:=14; m:=12;Int(x^l*(1-x)^m*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields Pi - 103993/33102

And also it works Lucas's formula for $\frac{104348}{33215}$

$$\frac{104348}{33215} - \pi = \frac{1}{38544}\int_{0}^{1}\frac{x^{12}(1-x)^{12}(1349-1060x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=7, i= -2409, j=4, k=1349, l=12, m=12$ - with regards to my above suggested generalization.

In Maple notation

i:=-2409; j:=4; k:=1349; l:=12; m:=12;Int(x^l*(1-x)^m*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 104348/33215 - Pi

And it works as well for $\frac{618669248999119}{196928538206400}$

$$\frac{618669248999119}{196928538206400} - \pi = \frac{1}{755216}\int_{0}^{1}\frac{x^{14}(1-x)^{12}(77159+124360x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=6, i= 47201, j=4, k=77159, l=14, m=12$ -with regards to my above suggested generalization.

In Maple notation

i:=47201; j:=4; k:=77159; l:=14; m:=12;Int(x^l*(1-x)^m*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

618669248999119/196928538206400 - Pi

I do not have computer math resources (Mathematica, Maple, etc.) to experimentally prove or disprove it for all larger n (but see my comment below).

Best Regards, Alexander R. Povolotsky


Matt B. in his answer to my question on Mathematics Stack Exchange has provided analytical proof and improved my parametric formula by reducing the number of parameters from 5 to 4 (see Seeking proof for the formula relating Pi with its convergents).

$$(-1)^n (\pi- \frac{p_n}{q_n}) = \int_0^1 \frac{x^{\epsilon+2m'}(1-x)^{2m'}(\alpha + \beta x^2) }{(\alpha - \beta) 2 ^{m'-2} (-1)^{\epsilon}(1+x^2)}dx$$.

Below is the list of parameters in Matt B.'s formula for all cases, covered in Stephen Lucas' publications - the stuff to the right of the arrow sign is the actual Maple code, which one could copy (while in the "edit" mode) and then paste into (let say) Inverse Symbolic Calculator (it accepts Maple code) and run it there.

NB I replaced for brevity some parameter names used by Matt B: "alpha" by "a", "beta" by "b", "epsilon" by "c", "m' " by "p".

104348/33215 - Pi -> a:=1349;b:=-1060;p:=6;c:=0;Int((x^(c+2*p)*(1-x)^(2*p)*(a+b*x^2))/((a-b)2^(p-2)((-1)^(c)*(1+x^2))),x=0...1)

Pi - 103993/33102 -> a:=124360;b:=77159;p:=6;c:=2;Int((x^(c+2*p)*(1-x)^(2*p)*(a+b*x^2))/((a-b)(2^(p-2))((-1)^(c)*(1+x^2))),x=0...1)

355/113 - Pi -> a:=25;b:=816;p:=4;c:=0;Int((x^(c+2*p)*(1-x)^(2*p)*(a+b*x^2))/((a-b)(2^(p-2))((-1)^(c)*(1+x^2))),x=0...1)

Pi - 333/106 -> a:=197;b:=462;p:=3;c:=-1;Int((x^(c+2*p)*(1-x)^(2*p)(a+b*x^2))/((a-b)(2^(p-2))((-1)^(c)*(1+x^2))),x=0...1)

22/7 - Pi -> a:=1;b:=0;p:=2;c:=0;Int((x^(c+2*p)*(1-x)^(2*p)*(a+b*x^2))/((a-b)(2^(p-2))((-1)^(c)*(1+x^2))),x=0...1)

Obviously parameters in the formula somehow depend on "n". The most straight forward dependency on "n" is observed in what Matt B named as "m'" (and I call it "p") : {2,3,4,4,6,6} ... Note that when "n" -> infinity, then the integral should come to 0 ...

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One also could check and see that my above generalization formula also applies to identities obtained by Jaume Oliver Lafont, described in the "Following Lucas (2009)" section in… – Alex Apr 8 '12 at 21:20

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